Test whether your observed defect or pass rate is significantly different from a target. Enter events, total trials, and your benchmark proportion to get z, p-value, and Wilson 95% CI.
Enter your events, trials, and target proportion, then press Calculate to run the test.
Your defect rate looks under the 5% target — but with a limited sample, can you be sure? Enter the lab to test it properly.
Provide the number of defectives and total items tested, plus the benchmark rate you're testing against — typically a quality target, contract limit, or historical average.
The calculator compares your actual rate (defectives ÷ total) to your benchmark, and measures how far they are apart. The bigger the gap, the smaller the p-value — and the stronger the evidence that the difference is genuine.
You get a 95% confidence interval — a range that's likely to contain the true rate — based on a method (Wilson's) that works well for all sample sizes. If that range sits entirely below your benchmark, you have clear proof of genuine improvement. The final decision compares your p-value to your chosen significance level.
Use the calculator above to test whether your observed pass or fail rate is genuinely above or below a benchmark. The 1 proportion z-test is the simplest hypothesis test for binary quality data and is the first check before claiming an improvement or flagging a supplier.
The 1 proportion z-test asks: could my observed proportion (p̂ = x/n) have come from a population where the true rate equals p₀? You compare the observed rate to a benchmark using a z-statistic. A significant result tells you the rate has genuinely shifted away from the target — either an improvement to claim or a problem to investigate.
This test asks whether your observed rate (defectives divided by total) is genuinely different from a target rate. It measures how many standard errors your observed proportion sits away from the target — the z-score — and converts that into a p-value. A p-value below 0.05 means the difference from target is real, not just sampling variation. The Wilson confidence interval gives a sensible range for the true rate, and it stays reliable even with small samples or very low defect counts.
Before a process change: 18 defects from 200 units (9%). After the change: 9 defects from 200 units (4.5%). That looks like an improvement — but with only 200 units, could it be luck?
The 1 Proportion Test gives p = 0.03. The improvement is real, not random variation in a small sample.
What to do with this: Lock in the process change — you have statistical evidence it works. Pair the p-value with the before/after numbers when making the business case to management.
Claiming an improvement without statistical evidence is one of the most common errors in quality management. The 1 proportion test stops teams from declaring victory based on a small favourable sample, and prevents rejecting a supplier based on a handful of defectives that could easily be sampling variation.
Use the 1 proportion test in the Control phase to verify that an improvement has genuinely moved the defect rate below the target. Also use it in supplier qualification to test whether a batch sample's defect rate meets the specification limit, and in audit sampling to confirm compliance with a maximum defect rate.
The 1 proportion test is the simplest entry point to hypothesis testing with attribute (binary) data in DMAIC. It feeds into the Analyse and Control phases, where you need to confirm that a baseline rate differs from a target and that improvements have been statistically verified before closing a project.
If the test is significant and your rate is below the target, document the result in your control plan as evidence of verified improvement. If not significant, calculate the sample size needed to achieve 80% power for the expected difference — then collect that sample before drawing conclusions. If the rate is above the target, investigate and escalate through the DMAIC Analyse phase to find the root cause.
Combine the 1 proportion test with sample size planning, control charts, and DMAIC project structure to verify and sustain defect-rate improvements.
Use the 1 proportion test when you have a single sample of pass/fail data and want to test whether the observed proportion is significantly different from a target or historical value (p₀). Examples: testing whether a process defect rate has genuinely fallen below the 5% target, whether a supplier's sample meets the 2% maximum defect rate, or whether an improvement has moved pass rates above 95%.
p₀ is the value you are testing against — the benchmark, target, or historical rate. Choose it based on your quality standard (e.g. a contractual defect limit), a historical baseline (e.g. last quarter's defect rate), or an industry benchmark. The test answers: 'Is our observed rate significantly different from p₀?'
The 95% Wilson confidence interval gives a range of plausible values for the true proportion. If the interval lies entirely below p₀, you can be confident the true rate is below the target. If it overlaps p₀, you cannot rule out that the true rate equals the target — possibly because the sample is too small to detect the difference with certainty.
Wilson's confidence interval (also called the score interval) has better coverage than the standard formula p̂ ± z√(p̂(1−p̂)/n), especially when p is close to 0 or 1 or the sample is small. The standard interval can produce impossible values below 0 or above 1, or have poor coverage with small samples. Wilson's interval is the modern recommended default for proportions.
A statistically verified result gives you the evidence to claim success or escalate a problem. Structured improvement training shows you how to build that case.
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